How good is a model if it can't predict a single positive class? [duplicate]

I have a training set of over a 100,000 points that is used to train a Logistic Regression Classifier (logit, since response is binary). The model is testing/fitted on a test set of 20,000 items. The test set is totally independent.

The ROC AUC value for this model is 0.85 which suggests that this is a good model. But I was not convinced. I picked a threshold $0.5$ (i.e., its classified positive if the model response $> 0.5$, negative if model response $< 0.5$).

At this threshold, I get the confusion matrix:

Confusion Matrix and Statistics

Reference
Prediction     0     1
0 33307   679
1     0     0

Accuracy : 0.98
95% CI : (0.9785, 0.9815)
No Information Rate : 0.98
P-Value [Acc > NIR] : 0.5102

Kappa : 0
Mcnemar's Test P-Value : <2e-16

Sensitivity : 0.00000
Specificity : 1.00000


So my question is, how good is the model if it is unable to predict a 'positive' class at 0.5 threshold?

My guess would be that the threshold of the model for labelling 'positive' is not $0.5$ in this case. Is this intuitive and make sense? Clearly the ROC AUC value is very high, which means that it does have a good TPR rate at lower thresholds.

• Why threshold 0.5 should be used? Why do not use estimated probabilities and take a decision at point where you get needed lift or if you have a profit function, at point of profit maximization? Commented Aug 20, 2015 at 4:15
• It may well be perfect. If the density of the minority class (weighted by it's prior probability) is lower everywhere than the weighted density of the positive class, then the Bayes optimal decision rule assigns all patterns to the majority class (see stats.stackexchange.com/questions/539638/…). If this is not an acceptable answer, it means the misclassification costs are not equal and you need to look at cost-sensitive learning (which can be implemented by altering the threshold for logistic regression). Commented Mar 6 at 18:14